Factorizations and Hardy–Rellich-type inequalities

Abstract

Dedicated with great pleasure to Helge Holden on the occasion of his 60th birthday.Abstract. The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter n-dimensional homogeneous scalar differential expressions T α,β := −∆ + α|x| −2 x · ∇ + β|x| −2 , α, β ∈ R, x ∈ R n \{0}, n ∈ N, n ≥ 2, and its formal adjoint, denoted by T + α,β , we show that nonnegativity of T + α,β T α,β on C ∞ 0 (R n \{0}) implies the fundamental inequality,

Factorizations and Hardy–Rellich inequalities on stratified groups

“…In this section, we obtain Hardy-Rellich and improved Hardy inequalities on stratified groups by factorization. We refer to [Ges84], [GP80] and to the recent paper [GL17] for obtaining Hardy and Hardy-Rellich inequalities using this factorization method in the isotropic abelian case.…”

“…Therefore, let us first recall known results in this direction. In the recent paper [GL17], inspiring our work, Gesztesy and Littlejohn used the nonnegativity of…”

“…)|x| −2 , for α, β ∈ R, x ∈ R n \{0} and n ≥ 2. As a result, they obtained the following inequality for all f ∈ C ∞ 0 (R n \{0}): We refer to [GL17] for a thorough discussion of the factorization method, its history and different features. We also refer to [GP80] for obtaining the Hardy inequality and to [Ges84] for logarithmic refinements by this factorization method.…”

Hardy and Hardy–Rellich type inequalities with Bessel pairs

“…This problem was investigated by many authors. See [1,6,9,10,13,14,15,20,21,22,23,27,28,36], to name just a few. See also the books [24,31] that are by now standard references on Hardy inequalities.…”

Estimates of Hardy - Rellich constants for polyharmonic operators and their generalizations

“…It should be noted that there are many works on Rellich inequalities for harmonic and polyharmonic operators as 2, Ω = R ∖ {0} and weight functions are the powers of | | (see [7]- [11] and the references therein). We shall consider the analogues of such inequalities as Ω ⊂ R is a bounded or an unbounded domain and the weight functions are the powers of dist( , Ω).…”